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The short answer is mold, tsunami's and earthquakes is why they didn't build with stone. There are few 'ancient' examples of stone building but most of them came from the Chinese style for the temples that were constructed and you can see the resemblance in the Chinese Pagoda vs. the Japanese toba. These Pagoda's were originally constructed as small shrines for a deceased Buddhist's ashes called a stupa and started to gain grandeur and were eventually made into Pagoda's. This is demonstrated in Nara's Hokki-ji (a mostly wooden structure) built during the Nara era it is the oldest Pagoda still in existence.
As for why they didn't build with stone; it goes back to mold primarily. Japan sits in a climate that is high in moisture and rarely drops below freezing in most of the inhabited spaces. You can see that mold, lichen, and moss build up and eat away at stone structures if not taken care of but the biggest problem is the mold making the people inside sick. Japanese used Cypress and Cryptomeria in their construction which like many other woods such as Cedar, White Oak and Teak repel mold and rot. The other portion to the equation is that you can also build your house off the ground slightly allowing for airflow underneath. This also helps reduce mold build up and reduces the amount of moisture in the structure.
This is in great contrast to the granite stone that was being used in the limited stone buildings in Japan. Granite is porous and will soak up and and hold moister making it more likely for mold to build up on it.
The other two reason why they didn't use stone like western cultures is because of tsunami's and earthquakes. As you can imagine, stone doesn't bend and twist like wood can so it shatters during an earthquake which Japan is prone to.
Hope this helps.
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